Optimal. Leaf size=104 \[ -\frac {a^8}{24 x^{24}}-\frac {8 a^7 b}{21 x^{21}}-\frac {14 a^6 b^2}{9 x^{18}}-\frac {56 a^5 b^3}{15 x^{15}}-\frac {35 a^4 b^4}{6 x^{12}}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{3 x^6}-\frac {8 a b^7}{3 x^3}+b^8 \log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} -\frac {a^8}{24 x^{24}}-\frac {8 a^7 b}{21 x^{21}}-\frac {14 a^6 b^2}{9 x^{18}}-\frac {56 a^5 b^3}{15 x^{15}}-\frac {35 a^4 b^4}{6 x^{12}}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{3 x^6}-\frac {8 a b^7}{3 x^3}+b^8 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^{25}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^9} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^8}{x^9}+\frac {8 a^7 b}{x^8}+\frac {28 a^6 b^2}{x^7}+\frac {56 a^5 b^3}{x^6}+\frac {70 a^4 b^4}{x^5}+\frac {56 a^3 b^5}{x^4}+\frac {28 a^2 b^6}{x^3}+\frac {8 a b^7}{x^2}+\frac {b^8}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^8}{24 x^{24}}-\frac {8 a^7 b}{21 x^{21}}-\frac {14 a^6 b^2}{9 x^{18}}-\frac {56 a^5 b^3}{15 x^{15}}-\frac {35 a^4 b^4}{6 x^{12}}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{3 x^6}-\frac {8 a b^7}{3 x^3}+b^8 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 104, normalized size = 1.00 \begin {gather*} -\frac {a^8}{24 x^{24}}-\frac {8 a^7 b}{21 x^{21}}-\frac {14 a^6 b^2}{9 x^{18}}-\frac {56 a^5 b^3}{15 x^{15}}-\frac {35 a^4 b^4}{6 x^{12}}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{3 x^6}-\frac {8 a b^7}{3 x^3}+b^8 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 89, normalized size = 0.86
method | result | size |
default | \(-\frac {a^{8}}{24 x^{24}}-\frac {8 a^{7} b}{21 x^{21}}-\frac {14 a^{6} b^{2}}{9 x^{18}}-\frac {56 a^{5} b^{3}}{15 x^{15}}-\frac {35 a^{4} b^{4}}{6 x^{12}}-\frac {56 a^{3} b^{5}}{9 x^{9}}-\frac {14 a^{2} b^{6}}{3 x^{6}}-\frac {8 a \,b^{7}}{3 x^{3}}+b^{8} \ln \left (x \right )\) | \(89\) |
norman | \(\frac {-\frac {1}{24} a^{8}-\frac {8}{3} a \,b^{7} x^{21}-\frac {14}{3} a^{2} b^{6} x^{18}-\frac {56}{9} a^{3} b^{5} x^{15}-\frac {35}{6} a^{4} b^{4} x^{12}-\frac {56}{15} a^{5} b^{3} x^{9}-\frac {14}{9} a^{6} b^{2} x^{6}-\frac {8}{21} a^{7} b \,x^{3}}{x^{24}}+b^{8} \ln \left (x \right )\) | \(91\) |
risch | \(\frac {-\frac {1}{24} a^{8}-\frac {8}{3} a \,b^{7} x^{21}-\frac {14}{3} a^{2} b^{6} x^{18}-\frac {56}{9} a^{3} b^{5} x^{15}-\frac {35}{6} a^{4} b^{4} x^{12}-\frac {56}{15} a^{5} b^{3} x^{9}-\frac {14}{9} a^{6} b^{2} x^{6}-\frac {8}{21} a^{7} b \,x^{3}}{x^{24}}+b^{8} \ln \left (x \right )\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 94, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, b^{8} \log \left (x^{3}\right ) - \frac {6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 94, normalized size = 0.90 \begin {gather*} \frac {2520 \, b^{8} x^{24} \log \left (x\right ) - 6720 \, a b^{7} x^{21} - 11760 \, a^{2} b^{6} x^{18} - 15680 \, a^{3} b^{5} x^{15} - 14700 \, a^{4} b^{4} x^{12} - 9408 \, a^{5} b^{3} x^{9} - 3920 \, a^{6} b^{2} x^{6} - 960 \, a^{7} b x^{3} - 105 \, a^{8}}{2520 \, x^{24}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.46, size = 97, normalized size = 0.93 \begin {gather*} b^{8} \log {\left (x \right )} + \frac {- 105 a^{8} - 960 a^{7} b x^{3} - 3920 a^{6} b^{2} x^{6} - 9408 a^{5} b^{3} x^{9} - 14700 a^{4} b^{4} x^{12} - 15680 a^{3} b^{5} x^{15} - 11760 a^{2} b^{6} x^{18} - 6720 a b^{7} x^{21}}{2520 x^{24}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.45, size = 100, normalized size = 0.96 \begin {gather*} b^{8} \log \left ({\left | x \right |}\right ) - \frac {2283 \, b^{8} x^{24} + 6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 91, normalized size = 0.88 \begin {gather*} b^8\,\ln \left (x\right )-\frac {\frac {a^8}{24}+\frac {8\,a^7\,b\,x^3}{21}+\frac {14\,a^6\,b^2\,x^6}{9}+\frac {56\,a^5\,b^3\,x^9}{15}+\frac {35\,a^4\,b^4\,x^{12}}{6}+\frac {56\,a^3\,b^5\,x^{15}}{9}+\frac {14\,a^2\,b^6\,x^{18}}{3}+\frac {8\,a\,b^7\,x^{21}}{3}}{x^{24}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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